The Parametrically Forced Pendulum Broer, Hendrik; Hoveijn, I.; Noort, M. van; Simó, C.; Vegter, Geert - PDF

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The Parametrically Forced Pendulum Broer, Hendrik; Hoveijn, I.; Noort, M. van; Simó, C.; Vegter, Geert Published in: Journal of dynamics and differential equations IMPORTANT NOTE: You are advised to consult

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The Parametrically Forced Pendulum Broer, Hendrik; Hoveijn, I.; Noort, M. van; Simó, C.; Vegter, Geert Published in: Journal of dynamics and differential equations IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2004 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Broer, H. W., Hoveijn, I., Noort, M. V., Simó, C., & Vegter, G. (2004). The Parametrically Forced Pendulum: A Case Study in 1½ Degree of Freedom. Journal of dynamics and differential equations, 16(4), Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: Journal of Dynamics and Differential Equations, Vol. 16, No. 4, October 2004 ( 2004) The Parametrically Forced Pendulum: A Case Study in 1 1 Degree of Freedom 2 H.W. Broer, I. Hoveijn, M. van Noort 1,C.Simó and G. Vegter Received October 31, 2002 This paper is concerned with the global coherent (i.e., non-chaotic) dynamics of the parametrically forced pendulum. The system is studied in a degree of freedom Hamiltonian setting with two parameters, where a spatio-temporal symmetry is taken into account. Our explorations are restricted to large regions of coherent dynamics in phase space and parameter plane. At any given parameter point we restrict to a bounded subset of phase space, using KAM theory to exclude an infinitely large region with rather trivial dynamics. In the absence of forcing the system is integrable. Analytical and numerical methods are used to study the dynamics in a parameter region away from integrability, where the analytic results of a perturbation analysis of the nearly integrable case are used as a starting point. We organize the dynamics by dividing the parameter plane in fundamental domains, guided by the linearized system at the upper and lower equilibria. Away from integrability some features of the nearly integrable coherent dynamics persist, while new bifurcations arise. On the other hand, the chaotic region increases. KEY WORDS: Hamiltonian dynamics; Bifurcations; Numerical methods; KAM theory Mathematical Subject Classification: 37J20, 37J40, 37M20, 70H INTRODUCTION We consider a parametrically forced pendulum in a Hamiltonian degree of freedom setting, given by the equation of motion ẍ = (α + β cos t)sin x. (1) Here x S 1 is the deviation from the upper equilibrium and ranges over the whole circle. The equation is given in inverted pendulum format, Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday /04/ / Springer Science+Business Media, Inc. 898 Broer, Hoveijn, van Noort, Simó and Vegter meaning that x = 0 corresponds to the upper equilibrium, where the pendulum stands up, while x = π denotes the lower equilibrium, where it hangs down. The independent variable is t S 1. The parameters α, β R correspond to the square of the eigenfrequency of the (free) pendulum at the lower equilibrium, and the amplitude of the forcing, respectively. Indeed, α = g/l, where g denotes the gravitational acceleration and l the length of the pendulum. Without loss of generality we restrict to α, β 0. This system is widely studied within the context of classical perturbation theory, that is, locally in several regions in phase and parameter space. Most notably, there is a long history of research of the Mathieu equation. This is the linearized equation of the forced pendulum at its equilibria, and it governs the stability of these points. See Fig. 1 for a stability diagram, also compare Mathieu [57], van der Pol and Strutt [71], Stoker [79], Meixner and Schäfke [65], Hale [39, 40], Weinstein and Keller [83,84], Levi et al. [50,56], and Broer et al. [18,24,25] b a Figure 1. Numerically computed stability diagram of the parametrically forced pendulum (1) on a large scale, revealing a checkerboard structure. On the curves going from the left towards the top side the stability of the upper equilibrium changes, on the curves coming from the α axis the lower equilibrium changes stability. Except near the diagonal α =β, every line consists, in fact, of two stability curves, that are extremely close together. Hence the stable regions of the upper equilibrium are extremely narrow, as are those of the lower equilibrium above the diagonal, while below this diagonal the unstable regions of the lower equilibrium are very small, compare [19]. The Parametrically Forced Pendulum 899 The non-linear system is nearly integrable for small β/α. This permits a local bifurcation analysis at the two equilibria, near the resonance points of the lower equilibrium on the α axis, see Broer and Vegter [28], and near the degenerate point (α, β) = (0, 0), see Broer et al. [12,13]. The existence of invariant tori can be shown using KAM theory, see Kolmogorov [47], Arnol d [4], and Moser [60]. Again the nearly integrable case of small β/α is a natural perturbative setting, but other settings are possible, e.g., see Moser [63, 64], Chierchia and Zehnder [30], and You [85], also compare Levi [51 53] for a discussion of a similar system Setting of the Problem and Sketch of the Results In contrast with the approaches sketched above, the present paper has a more global perspective, as it aims to explore the coherent (i.e., non-chaotic) dynamics of the forced pendulum, in dependence of the parameters (α, β) in the whole phase space and parameter plane. By coherent dynamics we mean all non-chaotic phenomena, that is, all periodic and quasi-periodic dynamics, and their bifurcations. Emphasis lies on orbits of low period, since these usually generate the largest stability islands, cf. [51]. This study combines perturbation theory and numerical tools, adopting the programme of Broer et al. [26]. Indeed, analytical results obtained in certain parameter regions serve as a framework and a starting point for numerical continuation to a larger part of parameter space. Furthermore, at representative parameter points phase portraits are computed. Such an interaction between analytical and numerical methods has proven to be a fruitful approach in the study of systems of this complexity. We obtain the following two main results. Firstly, at any parameter point (α, β) there exists a bounded region of interest such that outside this region the dynamics consists of a set of invariant tori of large measure, with thin layers of resonant dynamics in between. Secondly, the parameter plane can be divided in boxes, so-called fundamental domains. Theoretical and numerical evidence is presented that this structure organizes an important part of the coherent dynamics. Let us discuss these results in more detail. Our explorations mostly deal with the Poincaré map P of the forced pendulum, defined on the cylindrical section t = 0 mod 2πZ. This map has a spatial and a temporal symmetry, see Fig. 2 for some example phase portraits. Fig. 3 explains the coding of periodic points and bifurcations that will be used throughout this paper. For any parameter point (α, β), the coherent dynamics of P at sufficiently large y consists of invariant circles (of rotational type, i.e., 900 Broer, Hoveijn, van Noort, Simó and Vegter Figure 2. Phase portraits of the Poincaré mapatα =0.079, for various β, as indicated. Outside a region of interest, the dynamics mainly consists of invariant circles winding around the cylinder. Inside, the region of chaos increases with β, but coherent dynamics remains present. Both equilibria undergo period doubling and pitchfork bifurcations. Parts of invariant manifolds of (unstable) equilibria are also plotted. Invariant manifolds and periodic points born at the equilibria are marked according to the coding in Fig. 3. In some diagrams there are unmarked stable fixed points above and below the lower equilibrium, corresponding to resonant orbits of the vector field X with frequency 1 in the x-direction. In the bottom left diagram, these fixed points have destabilized in a period doubling bifurcation. For more details, we refer to Section 1.1. winding around the cylindrical phase space) with confined chaotic motion and strings of islands in between. We prove by KAM theory that the measure of the set of invariant circles is exponentially close to full measure as y. This region of invariant circles and small islands therefore is excluded from the analysis, thereby restricting to a region of interest in The Parametrically Forced Pendulum 901 Figure 3. Coding of periodic points, invariant manifolds of equilibria, and bifurcations of the Poincaré map, used in phase portraits, bifurcation and stability diagrams. phase space. We note that for the free pendulum this coincides with the region between the separatrices. A very accurate numerical estimate on a large domain in the parameter plane, supported by asymptotic theoretical estimates, shows that the region of interest is bounded by y ±2 α ± β for all sufficiently large (α, β) R 2 0. For a rigorous bound, using KAM theory, we refer to [21]. In the region of interest, we organize our explorations by dividing the parameter plane into so-called fundamental domains, based on the stability types of the upper and lower equilibria. Fig. 4 shows the corresponding stability diagrams in the (α, β) plane, where shading indicates stability. The stability boundaries are curves of pitchfork (PF) or period doubling (PD) bifurcations, as indicated, where the pitchfork bifurcation has codimension one due to a spatial symmetry of the system, that will be explained below. Combining the two stability diagrams we see that the parameter plane consists of regions of four different types, depending on the stability types of the two equilibria. These regions are collected mostly in groups of four, one of each type, forming what will be called fundamental domains, again see Fig. 4, also compare Fig. 5 for a sketch of one domain. The boundaries of these domains are formed by stability curves. Most fundamental domains belong to four types, depending on the bifurcation type of their boundaries. Indeed, these domains have four sides, where opposite sides correspond to different bifurcation types, either PF or PD, leading to a total of four possibilities. The fundamental domains mostly appear in blocks of 2 2 domains, one of each type, again see Fig. 5. Moving across such a block, both 902 Broer, Hoveijn, van Noort, Simó and Vegter Figure 4. Part of the (numerically computed) stability diagrams of the equilibria of the pendulum (1), first separate, then combined. Shading indicates stability. The diagrams are symmetric around the α axis. At curves labeled PF or PD pitchfork bifurcations or period doubling bifurcations take place, respectively. In the rightmost diagram, thickened curves correspond to boundaries of fundamental domains. Figure 5. Left: Sketch of one fundamental domain, with an indication of the stability types and bifurcations of the two equilibria. The bifurcations on the stability boundaries depend on the type of fundamental domain. In each region the upper equilibrium is in the top phase portrait, and the lower in the bottom one. Right: Sketch of one block of 2 2 fundamental domains (bounded by thick lines), where the bifurcation types of the stability boundaries are indicated. horizontally and vertically, one intersects two stability boundaries of PF type (where the trace of the linearized Poincaré map at the relevant equilibrium equals 2), two of PD type (where the trace is 2), and finally another one of PF type, in that order. Part of the coherent dynamics of the nonlinear system is governed by the fundamental domains. Indeed, for small β/α, normal form theory yields an integrable approximation to P, valid near the two equilibria, cf. [12, 13, 28]. This is continued numerically to larger parameter values. The approximation shows that the stability types and bifurcations of the upper and lower equilibria (and hence the appearance of stable regions) are the same in fundamental domains of the same type. We note that below the diagonal α = β there is a single row of fundamental domains. The lower The Parametrically Forced Pendulum 903 equilibrium is mostly stable here, and hence this region contains the largest part of the coherent, near integrable dynamics of the full problem. Away from integrability we find other local bifurcations that involve large stable domains in phase space. These are also investigated by numerical means, and it turns out that their codimension one bifurcation curves follow the fundamental domain structure in the parameter plane. This leads us to the conjecture that the fundamental domains organize a substantial part of the global coherent dynamics. Remarks: 1. A Fig. 2 illustrates, the islands between the rotational invariant circles (outside the region of interest ) are very hard to detect, since the largest ones are close to the chaotic domain. As an example, we have plotted a period 4 island in the top right diagram passing through (x, y) = (0.0915) (indicated by the arrow). This island intersects the line x = 0 between y = and y = 0.926, and goes horizontally from 0.14 to 0.14, where the related hyperbolic period 4 points are located. 2. For most (α, β) the region of interest also includes orbits that in the integrable case β = 0 lie outside this region. For example, the large stable domains above and below the lower equilibrium in the phase portraits of Fig. 2 lie inside the region of interest, but correspond to a resonant invariant circle with rotation number one in the integrable system Acknowledgments We thank Boele Braaksma, Alan Champneys, Francesco Fassò, Àngel Jorba, Bernd Krauskopf, and Evgeny Verbitski for valuable discussion during the preparation of this paper. The research of the fourth author has been partially supported by grants BFM C02-01 (Spain), 2001SGR- 70 (Catalonia) and INTAS The computer clusters (HIDRA and EIXAM) of the UB-UPC Dynamical Systems Group have been widely used Overview Let us give a short overview of the contents of this paper. In Section 2 the system is introduced and some properties regarding symmetries and linearization are discussed. Section 3 applies KAM theory to show persistence of Diophantine invariant circles of P, and gives estimates on the measure of the union of these circles and on the size of the region 904 Broer, Hoveijn, van Noort, Simó and Vegter of interest. Adiabatic estimates on this size are given in Appendix A. In Section 4 the Poincaré map P is studied in several fundamental domains. 2. PRELIMINARIES In this section some properties of the system, like symmetries, are discussed. The system (1) of the parametrically forced pendulum can be written as a vector field X(x,y,t; α, β) = t + y x (α + βρ(t))v (x) y, (2) where y =ẋ denotes velocity as before, (x,y,t) S 1 R S 1, ρ(t) = cos t,v(x) = cos x 1, and (α, β) R 2. This vector field is Hamiltonian, with time-dependent Hamilton function H(x,y,t; α, β) = 1 2 y2 + (α + βρ(t))v (x). (3) The vector field X has several symmetries. Since V is even, it has a spatial symmetry given by S : (x,y,t) (-x,-y,t). This means that S X = X, and X is called S-equivariant. Furthermore, X is R-reversible, meaning that R X = X, where the temporal symmetry R : (x,y,t) ( x, y,t) is due to the evenness of ρ. There are other symmetries, involving both phase and parameter space. Indeed, since V(x)= V(x+ π),x is T -equivariant, where T :(x,y,t; α, β) (x +π,y,t; α, β). Finally, X is U-equivariant, with U : (x,y,t; α, β) (x,y,t+ π; α, β), because ρ(t)= ρ(t + π). By these two symmetries without loss of generality we can restrict to the first quadrant α, β 0 of the parameter plane. Since the system is 2π-periodic in time, it is natural to consider its Poincaré or stroboscopic map P on the section t =0, corresponding to the flow over time 2π of X. The map P is defined implicitly by (P (x, y), 2π)= X 2π (x, y, 0), where X t is the flow over time t of X. Since X is divergence free, P is area and orientation preserving. Moreover it inherits the symmetries of X. This means that P is S-equivariant and R-reversible, that is, SPS = P and RPR = P 1, where S : (x, y) ( x, y) and R : (x, y) (x, y). The symmetries T and U also carry over to symmetries of P in a trivial way. The stability of the two equilibria is determined by the linearized system, given by the so-called Mathieu equation ẍ = (α + βρ(t))x. The Parametrically Forced Pendulum 905 The Poincaré map of the Mathieu equation equals the linearization of the map P at the upper equilibrium. Thus the Mathieu equation at the parameter point (α, β) determines the stability of the upper equilibrium (x,y,t)= (0, 0,t) at (α, β), and, by the symmetry T of the nonlinear system, the stability of the lower equilibrium (x,y,t)= (π, 0,t),at( α, β). Hence, combining the stability diagram of the Mathieu equation with a copy rotated around the origin over angle π results in a diagram where each parameter point shows the stability type of both equilibria, resulting in the checkerboard of fundamental domains shown in Figs. 1 and 4. (Because of the symmetry U the copy can also be reflected in the β-axis instead of rotated.) We introduce a codification of the checkerboard. Each fundamental domain can be identified by a pair (column number, row number), starting with (1,1), as illustrated in Fig. 6. The stability regions in a fundamental domain can be identified by a third label equal to one of the strings UU, US, SU, SS, where the former (latter) letter determines the stability of the lower (upper) equilibrium; U means unstable and S stable. Figure 6. The stability diagram of the parametrically forced pendulum is a checkerboard of fundamental domains, that mostly consist of four stability regions. The domains are bounded by thickened (black) curves and codified as indicated. In the shaded regions the lower or upper equilibrium is stable. (In many domains these regions are too narrow to be seen). 906 Broer, Hoveijn, van Noort, Simó and Vegter Remark. In [12,13,28] more general cases of V and ρ are considered, where the symmetries S and R are optional, while T and U play no role. In the present paper we restrict to the simplest case with maximal symmetry, and take the potential function V(x)= cos x 1 of the pendulum and the forcing function ρ(t)= cos t of the classical Mathieu case. 3. INVARIANT CIRCLES AND THE REGION OF INTEREST It is well known from KAM theory that for y sufficiently large and satisfying a Diophantine condition, the dynamics of P is almost completely quasi-periodic, compare Moser [63, 64], Chierchia and Zehnder [30], and You [85]. In between these invariant circles of quasi-periodic motion one generically expects confined chaotic motion and strings of islands, also called Poincaré-Birkhoff chains. Indeed, the Poincaré-Birkhoff theorem [16, 17] implies that in between any two invariant circles there exist periodic points of all intermediate rational rotation numbers. In this section we obtain the following results. At any given parameter point (α, β), invariant circles of P with Diophantine rotation number exist for y C(α, β), forming a Whitney smooth Cantor foliation of phase space. Here C 0 is some increasing
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